*2.3. Similarity Solutions*

In this work, the subsequent similarity transformation is introduced (Roy et al. [55])

$$\eta = \left(\frac{a}{\nu\_f(1-bt)}\right)^{1/2} y, \; \psi = \left(\frac{av\_f}{1-bt}\right)^{1/2} x \, f(\eta), \; N = \left(\frac{a}{\nu\_f(1-bt)}\right)^{1/2} \frac{a}{(1-bt)} x \, h(\eta), \; \theta(\eta) = \frac{T-T\_{\infty}}{T\_w - T\_{\infty}} \tag{10}$$

where *νf* and *η* are the fluid kinematic viscosity and similarity variable, while *f* , *h* and *θ* are the dimensionless function. Further, primes signify the differentiation with respect to *η*, while the stream function *ψ* is specified as *v* = −*∂ψ*/*∂<sup>x</sup>* and *u* = *∂ψ*/*∂y*.

Invoking the similarity variables (10), Equation (1) is identically fulfilled and Equations (2), (3) and (9) are reduced into the following similarity equations

$$\frac{\mu\_{\text{lnf}}/\mu\_f}{\rho\_{\text{lnf}}/\rho\_f}(1+K)f^{\prime\prime} - f^{\prime2} + f \, f^{\prime\prime} + 1 - A \left(f^{\prime} - 1 + \frac{1}{2}\eta f^{\prime\prime}\right) + \frac{K}{\rho\_{\text{lnf}}/\rho\_f} h^{\prime} = 0 \tag{11}$$

$$\frac{1}{\rho\_{\rm Imf}/\rho\_f} \left( \frac{\mu\_{\rm Imf}}{\mu\_f} + \frac{K}{2} \right) h'' + fh' - f'h - \frac{A}{2} \left( 3h + \eta h' \right) - \frac{K}{\rho\_{\rm Imf}/\rho\_f} \left( 2h + f'' \right) = 0 \tag{12}$$

$$\frac{1}{\Pr(\rho \mathbb{C}\_p)\_{\text{hnf}} / \left(\rho \mathbb{C}\_p\right)\_f} \left(\frac{k\_{\text{hnf}}}{k\_f} + \frac{4}{3}Rd\right)\theta'' + f\theta' - 2f'\theta - \frac{A}{2}\left(3\theta + \eta\theta'\right) = 0 \tag{13}$$

Here, the material parameter, unsteady parameter, Prandtl number and radiation parameter which denoted by *K*, *A*, Pr and *Rd* are defined by

$$K = \frac{\kappa}{\mu\_f}, \ A = \frac{b}{a'} \Pr = \frac{\nu\_f}{\alpha\_f}, \ Rd = \frac{4\sigma^\* T\_\infty^3}{k\_f k^\*},\tag{14}$$

The conditions (6) become

$$\begin{array}{ccccc} f'(0) = \mathfrak{c}/a = \lambda, & f(0) = 0, & h(0) = -nf''(0), & \theta(0) = 1, \\ \qquad f'(\eta) \to 1, & h(\eta) \to 0, & \theta(\eta) \to 0 \quad \text{as} \quad \eta \to \infty \end{array} \tag{15}$$

where the stretching/shrinking parameter is denoted by *λ* with *λ* > 0 signifies the sheet is stretch, *λ* = 0 refers to static plate and *λ* < 0 denotes the sheet is shrunk.

In this investigation, the physical quantities of interest are specified as

$$\mathbf{C}\_{f} = \frac{1}{\rho\_{f} u\_{\varepsilon}^{2}} \left[ \left( \mu\_{\mathrm{fuf}} + \mathbf{x} \right) \left( \frac{\partial \mathbf{u}}{\partial y} \right) + \kappa \mathbf{N} \right]\_{y=0},\\\mathrm{Nu}\_{x} = \frac{\mathbf{x}}{k\_{f} (T\_{\mathrm{w}} - T\_{\infty})} \left[ -k\_{\mathrm{fm}} \left( \frac{\partial T}{\partial y} \right)\_{y=0} + q\_{r}|\_{y=0} \right] \tag{16}$$

here, *Cf* is the skin friction coefficient and *Nux* is the Nusselt number. Using variables (10) and (16), the following local skin friction coefficient and local Nusselt number (heat transfer rate) are achieved

$$\mathcal{C}\_f \text{Re}\_x^{1/2} = \left(\frac{\mu\_{\text{lnf}}}{\mu\_f} + K\right) f''(0) + Kh(0), \text{ Nu}\_x \text{Re}\_x^{-1/2} = -\left(\frac{k\_{\text{lnf}}}{k\_f} + \frac{4}{3} Rd\right) \theta'(0) \tag{17}$$

where Re*x* = *uex*/*<sup>ν</sup>f* is the local Reynolds number.
